This is an adaptation of the official lasagne from the Bolognese Chamber of Commerce. I've made a couple of changes suggested to me by people from the area, such as which meat to use or if there should be milk added or not. One thing never in question, though, is the green lasagne sheets. These are a must. The ragu is an authentic sauce, simple without garlic or herbs and spices. Using fresh pasta keeps you from having to pre-boil it.
Saute pancetta in a large stockpot over medium-high heat until fat has rendered, 1 to 2 minutes. Add carrot, celery, onion, and oil. Cook and stir until vegetables have softened and onion is translucent, about 5 minutes. Add beef and lamb. Cook, stirring and mashing meat into small crumbles, until browned, 5 to 7 minutes more.
Stir apple juice into the stockpot and bring to a boil. Add tomato sauce and stir; pour in 1 cup broth. Reduce heat to low and partially cover the pot to allow steam to escape. Simmer sauce for 1 hour, stirring occasionally.
Pour 1 more cup of broth into the sauce. Continue simmering until thick, 1 to 1 1/2 hours. Add more broth only if absolutely necessary; ragu should drop, not run, off a spoon.
Start preparing pasta after sauce has been cooking for 1 1/2 hours. Bring water and salt to a boil in a small saucepan. Add spinach and cover until it returns to a boil. Stir gently and cook until spinach is heated through. Remove from heat, leave covered, and let cool.
Drain spinach in a mesh strainer set over a bowl, pressing it down with a fork to squeeze out as much water and possible; reserve the liquid. Puree spinach in a blender until smooth.
Sift flour into a bowl. Make a well in the center; add eggs, spinach, and about 1 tablespoon of the reserved liquid. Mix together by hand or using the dough hook of a stand mixer, adding more liquid if needed. Knead until dough becomes a smooth ball. Cover with plastic wrap and let sit at room temperature for 30 minutes.
Taste the ragu and adjust salt. Remove from heat and add milk; stir well and set aside to cool.
Divide pasta dough into 4 equal sections. Use a pasta machine, or rolling pin, dusted with flour, to roll pasta into 1/16-inch thick sheets. Lay sheets on clean dish towels to help them dry. Let rest for 20 minutes.
Meanwhile, melt butter in a saucepan over medium heat. Do not allow it to brown or burn. Add flour and whisk constantly until golden. Add 1/2 cup milk, whisking constantly. Repeat with remaining milk. Simmer until bechamel is thickened, about 10 minutes. Season with nutmeg and salt.
Preheat the oven to 350 degrees F (175 degrees C).
Spread a bit of the ragu sauce over the base of a 9x13-inch baking dish. Add a layer of pasta. Top with a spoonful of sauce and bechamel. Add a spoonful of grated Parmigiano-Reggiano. Top with another layer of pasta. Add more sauce, bechamel, and cheese. Repeat layers until you reach the top; finish with a layer of pasta, bechamel, and cheese.
Bake in the preheated oven until the sauce is bubbling and the cheese on top has browned, 25 to 30 minutes. Let lasagne rest for 10 or 15 minutes at room temperature before slicing.